Math 641-600 Final Exam Review (Fall 2019)
The final exam will be given on Tuesday, Dec. 10, 8-10 am, in our
usual classroom. It will cover sections 2.2.4 (DFT — see notes),
2.2.7, 3.2 - 3.6, 4.1 - 4.3.2, 4.5, and all class notes, starting from
the discrete Fourier, except for the notes on X-ray tomography. The
test will consist of the following: statements and/or proofs or
sketches of proofs of theorems; statements of definitions; proofs of
short propositions or solutions of problems similar to ones done in
the
homework or in class. Office hours: Wednesday (12/4),
11:30-12:30; Friday (12/6), 12:30-3:30; and Monday, 11-12:30. For
other times, send me an email to arrange an appointment. (The office
hours are subject to change. If I need to do that, I'll email the
class.)
The Discrete Fourier Transform and Finite Elements
Operators and integral equations
- Bounded operators (§3.2, Bounded
Operators & Closed Subspaces,
and
Projection theorem, the Riesz representation theorem, etc.)
- Norms of linear operators, unbounded operators, continuous linear
functionals, spaces associated with operators
- Hilbert-Schmidt kernels
- The Projection Theorem
- The Riesz Representation Theorem
- Existence of adjoints of bounded operators
- Fredholm alternative
- Compact operators (§3.3, §3.5, Compact
Operators and on
Closed Range Theorem.)
- Finite rank operators, $\mathcal C(\mathcal H)$ is a closed
subspace of $\mathcal B(\mathcal H)$ (Keener, Theorem 3.4 and Compact
Operators, Theorem 2.6) and Hilbert-Schmidt kernels/operators
- Closed Range Theorem, Fredholm alternative, resolvents and
kernels
- Spectral theory for compact, self-adjoint operators, K = K*
(§3.4,
Spectral Theory for Compact Operators.)
- Eigenvalues and eigenspaces
- Eigenvalues are real; eigenvectors for distinct eigenvalues are
orthogonal
- Eigenspaces are finite dimensional
- The only limit point of the set of eigenvalues is 0.
- "Maximum principle" (Keener, p. 117 and Spectral
Theory for Compact Operators, Lemma 2.5)
- The spectral theorem (Keener, Theorem 3.6 and Spectral
Theory for Compact Operators, Theorem 2.7)
- Solving eigenvalue problems
- Contraction Mapping Theorem, Neumann series (Keener, §3.6)
Distributions and differential operators
- Test function space $\mathcal D$, distribution space $\mathcal
D'$, integral representation; delta functions, derivatives of
distributions (§4.1, Example
problems on distributions.)
- Green's functions for 2nd order operators (§4.2)
- Domain of an operator, adjoints of 2nd order operators, domain of
the adjoint, self-adjoint differential operators (§4.3)
- Completeness of the set of o.n. eigenfunctions for a self-adjoint
differential operator (class notes and Keener, §4.5)
Updated 12/1/2019 (fjn).